Nonlinear response theory

Maciejko J, Wang J, Guo H (2006) Time-dependent quantum transport far from equilibriam an exact nonlinear response theory. Phys Rev B 74 085324... 

D.C. Langreth, in J.T. Devreese, V.E. van Doren (Eds.), Linear and nonlinear response theory with applications, Linear and Nonlinear Electron Transport in Solids, Plenum Press, New York, 1975, p. 3. 

D. Evans and G. Morriss, Rhys. Rev. A., 30, 1528 (1984). Nonlinear-Response Theory for Steady Planar Couette Flow. 

Although the full time-dependent electron density contains much useful information, it is hard and expensive to compute, and not needed for many basic properties. Most applications in chemistry therefore use only linear (or nonlinear) response theory to obtain the first-order (or higher-order) change in the electron density due to a time-dependent electric perturbation. Most properties of interest (such as excitation energies and oscillator strengths) can be obtained from the linear response equations. 

Evans, D.J. and Morriss, G.P. (1984) Nonlinear-response theory for steady planar Couette flow. Phys. Rev. A, 30, 1528-1530. 

The most recent contribution to the field of nonlinear response theory comes from Keyes, Space, and collaborators [56,57]. Their approach results in an exact classical response function written in terms of classical time correlation functions (TCF). The response takes into account the nonlinear polarizability and is used in a fully anharmonic MD simulation to simulate the fifth-order response of CS2 (see Fig. 1.14). The results are strikingly similar to those obtained by Jansen et al. as well as to the simulations, both MD and adiabatic INM, of liquid Xe (see Figs. 1.8 and 1.10). The dominant features are the ridge along the probe delay and the distinct lack of signal along the pump delay. 

Flaving now developed some of the basic notions for the macroscopic theory of nonlinear optics, we would like to discuss how the microscopic treatment of the nonlinear response of a material is handled. Wliile the classical nonlinear... 

The focus of the present chapter is the application of second-order nonlinear optics to probe surfaces and interfaces. In this section, we outline the phenomenological or macroscopic theory of SHG and SFG at the interface of centrosymmetric media. This situation corresponds, as discussed previously, to one in which the relevant nonlinear response is forbidden in the bulk media, but allowed at the interface. 

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

In recent years, these methods have been greatly expanded and have reached a degree of reliability where they now offer some of the most accurate tools for studying excited and ionized states. In particular, the use of time-dependent variational principles have allowed the much more rigorous development of equations for energy differences and nonlinear response properties [81]. In addition, the extension of the EOM theory to include coupled-cluster reference fiuictioiis [ ] now allows one to compute excitation and ionization energies using some of the most accurate ab initio tools. 

As mentioned, this equivalence is a consequence of the fluctuation-dissipation theorem (the general basis of linear response theory [51]). In (12.68), we have dropped nonlinear terms and we have not indicated for which state Variance (rj) is computed (because the reactant and product state results only differ by nonlinear terms). We see that A A, AAstat, and AAr x are all linked and are all sensitive to the model parameters, with different computational routes giving a different sensitivity for AArtx. 

Neiss, C., Hattig, C. Frequency-dependent nonlinear optical properties with explicitly correlated coupled-cluster response theory using the CCSD(R12) model. J. Chem. Phys. 2007, 126, 154101. 

The important step of identifying the explicit dynamical motivation for employing centroid variables has thus been accomplished. It has proven possible to formally define their time evolution ( trajectories ) and to establish that the time correlations ofthese trajectories are exactly related to the Kubo-transformed time correlation function in the case that the operator 6 is a linear function of position and momentum. (Note that A may be a general operator.) The generalization of this concept to the case of nonlinear operators B has also recently been accomplished, but this topic is more complicated so the reader is left to study that work if so desired. Furthermore, by a generalization of linear response theory it is also possible to extract certain observables such as rate constants even if the operator 6 is linear. 

It is noted that the derivation of this equation involves the phenomenological concept of the nonlinear response of the atoms. This equation is derived on the basis of the standard Abelian theory of electromagnetism, which is linear, and where the nonlinearity obtains by imposing nonlinear material responses. The physical underpinnings of these nonlinearities are not completely described. This soliton wave corresponds to diphotons, or photon bunches. 

A number of approaches have been described in the literature for calculating the microscopic nonlinear response and are reviewed in detail elsewhere (1). In this section, these methods are briefly mentioned and a simplified version of time dependent perturbation theory is used to illustrate the intuitive aspects of the microscopic nonlinear polarization. 

Several other studies have appeared that are worthy of note. In a series of works by Keller [89-94] and Apell [95], the nonlocal nonlinear response for free-elec-tron-like metals have been examined using various theoretical approaches which are basically extensions of linear theories on the optical response of metals. The results [92] reduce to those obtained by Rudnick and Stern [26] using a similar approach when the free-electron gas is considered to be homogeneous. 

All the linear and nonlinear optical properties introduced above are therefore expressed in terms of linear, quadratic and cubic response functions. They can be computed with high efficiency using analytical response theory [9] with a variety of electronic structure models [8],... 

The incentive and the main goal of this section are to consistently extend the conventional theory on the case of a nonlinear response and by that to confirm its validity. While doing that we propose practical schemes (both exact and approximate) to handle linear and cubic dynamic responses in the framework of classical superparamagnetism. Applying our results to the reported data on the nonlinear susceptibility of Cu-Co precipitates, we demonstrate that a fairly good agreement may be achieved easily. 

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